An experiment stain of bacteria is being used to treat an oil spill. A colony of 2^13 cells will be sufficient to ocean up the spill. If the colony is started with a single cell, and cells increase in number by splitting every 15 minutes, in how many hours will the colony reach the desired size?
the colony doubles 4 times every hour. So, after t hours, it has reached 2^(4t) cells. Clearly 4t = 13
3.5 hours
To calculate the time it will take for the colony to reach the desired size, we can use the formula:
N = N0 * 2^(t/Δt)
Where:
- N is the total number of cells in the colony after time t
- N0 is the initial number of cells in the colony (1 cell)
- t is the time in minutes
- Δt is the time it takes for each cell to split (15 minutes)
We need to find the value of t (in minutes) when N equals 2^13.
2^13 = 1 * 2^(t/15)
To solve for t, we can take the logarithm base 2 of both sides:
13 = (t/15) * log2(2)
Since log2(2) simplifies to 1, we have:
13 = t/15
Multiplying both sides by 15:
t = 13 * 15
t = 195 minutes
To convert this to hours, divide by 60:
t = 195 / 60
t ≈ 3.25 hours
Therefore, it will take approximately 3.25 hours for the colony to reach the desired size.
To solve this problem, we need to determine how many times the single bacterial cell will divide until we reach a colony size of 2^13 cells.
Since each cell divides into two cells every 15 minutes, the number of cells will double every 15 minutes. So, after 15 minutes, we'll have 2 cells; after 30 minutes, we'll have 4 cells, and so on.
We can calculate the number of times the single cell divides by finding the exponent that gives us 2^13. In other words, we need to solve the equation 2^n = 2^13.
By comparing the exponents, we can see that n = 13.
Therefore, the single bacterial cell will divide 13 times until we reach a colony size of 2^13 cells.
Since each division takes 15 minutes, we can multiply the number of divisions by 15 to find the total time in minutes. So, the colony will reach the desired size in 13 * 15 = 195 minutes.
To convert this time into hours, we divide by 60 (since there are 60 minutes in an hour). Thus, the colony will reach the desired size in 195 / 60 = 3.25 hours.